for some spinors \(\alpha ^{A}\) , \(\mu ^{A^{\prime }}\) , depending on whether the 2-form \(g(N\cdot ,\cdot )\)  is SD or ASD.
We may refer to the pair \((g,N)\)  as an `almost-null Kähler structure'.
It follows that the space of such maps at a point \(x\in M\)  is simply the space of spinors at \(x\) .
If the spinor \(\alpha ^{A}\)  (or \(\mu ^{A^{\prime }}\) ) is parallel under the Levi-Civita connection of \(g\) ,
then this is called a Null-Kähler structure, see [1]} and [2]}.
(If \(N\)  is real, then the signature of \(g\)  must be split, however here we allow \(N\)  to be complex.)

